Properties

Label 1-4235-4235.1082-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.624 + 0.780i$
Analytic cond. $455.113$
Root an. cond. $455.113$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.603 + 0.797i)2-s + (0.994 + 0.104i)3-s + (−0.272 + 0.962i)4-s + (0.516 + 0.856i)6-s + (−0.931 + 0.362i)8-s + (0.978 + 0.207i)9-s + (−0.371 + 0.928i)12-s + (−0.441 − 0.897i)13-s + (−0.851 − 0.524i)16-s + (−0.976 − 0.217i)17-s + (0.424 + 0.905i)18-s + (−0.749 − 0.662i)19-s + (−0.971 − 0.235i)23-s + (−0.964 + 0.263i)24-s + (0.449 − 0.893i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.603 + 0.797i)2-s + (0.994 + 0.104i)3-s + (−0.272 + 0.962i)4-s + (0.516 + 0.856i)6-s + (−0.931 + 0.362i)8-s + (0.978 + 0.207i)9-s + (−0.371 + 0.928i)12-s + (−0.441 − 0.897i)13-s + (−0.851 − 0.524i)16-s + (−0.976 − 0.217i)17-s + (0.424 + 0.905i)18-s + (−0.749 − 0.662i)19-s + (−0.971 − 0.235i)23-s + (−0.964 + 0.263i)24-s + (0.449 − 0.893i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.624 + 0.780i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1082, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.624 + 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.644559900 + 3.420834249i\)
\(L(\frac12)\) \(\approx\) \(1.644559900 + 3.420834249i\)
\(L(1)\) \(\approx\) \(1.524032237 + 0.9329943309i\)
\(L(1)\) \(\approx\) \(1.524032237 + 0.9329943309i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.603 + 0.797i)T \)
3 \( 1 + (0.994 + 0.104i)T \)
13 \( 1 + (-0.441 - 0.897i)T \)
17 \( 1 + (-0.976 - 0.217i)T \)
19 \( 1 + (-0.749 - 0.662i)T \)
23 \( 1 + (-0.971 - 0.235i)T \)
29 \( 1 + (0.870 + 0.491i)T \)
31 \( 1 + (0.640 + 0.768i)T \)
37 \( 1 + (-0.0380 + 0.999i)T \)
41 \( 1 + (0.974 - 0.226i)T \)
43 \( 1 + (0.540 - 0.841i)T \)
47 \( 1 + (0.424 - 0.905i)T \)
53 \( 1 + (0.524 + 0.851i)T \)
59 \( 1 + (0.290 + 0.956i)T \)
61 \( 1 + (0.123 - 0.992i)T \)
67 \( 1 + (-0.189 + 0.981i)T \)
71 \( 1 + (0.993 - 0.113i)T \)
73 \( 1 + (-0.132 + 0.991i)T \)
79 \( 1 + (0.935 - 0.353i)T \)
83 \( 1 + (0.825 + 0.564i)T \)
89 \( 1 + (-0.580 - 0.814i)T \)
97 \( 1 + (-0.967 - 0.254i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.10974914892256788404268188566, −17.59782178163132516026896919223, −16.388209435407122157148048014535, −15.69771788292127683302474297787, −15.01526852489069178204404722253, −14.371910285745017502220744039347, −13.912006982425239792537226095723, −13.24515127569041464142884810770, −12.548753408680579027414294271686, −12.00471267384525433722978731880, −11.13771152035290942512657533911, −10.418088363196382544383063664269, −9.63099772395260047467964941877, −9.22109701649131932097405244040, −8.35226061164436430824186768153, −7.64567264812222292501065366039, −6.53146510115400611043965707901, −6.16144644949245560464811162881, −4.909890349818719812695152415145, −4.07996915268007988177166659080, −3.94609266791975309884725570964, −2.62853698905744857767063864442, −2.26690099219167069647672294187, −1.53050333105678787801000891201, −0.42903870572718641760572384854, 0.7716839261733073594794120059, 2.29099600899138698450043827691, 2.66566537849853269851590527263, 3.57643897427625868037866685389, 4.36845731988020699965700188248, 4.86247912015445053029540177589, 5.81905668933484762771498981367, 6.76878632875220787049875580407, 7.17280584808959814384399975172, 8.146404423951736475942064051256, 8.519849636295888984669991628027, 9.19887099755195117497176264872, 10.12313290744467998979855775625, 10.79477134040836110242529284000, 11.96335995879422245208939965577, 12.56075911538273063544833833872, 13.19994785787036143355238500888, 13.86559712353471238111411244461, 14.30254928312227895471691934902, 15.32284130285269150528314352251, 15.38407648657738601119571349756, 16.13746595019261143979626567739, 17.017393499363477405244829991019, 17.770143331953664900385494318490, 18.20637012790547929526443973724

Graph of the $Z$-function along the critical line